[Show abstract][Hide abstract] ABSTRACT: Equations have been developed to describe cardiac action potentials and pacemaker activity. The model takes account of extensive developments in experimental work since the formulation of the M.N.T. (R. E. McAllister, D. Noble and R. W. Tsien, J. Physiol., Lond. 251, 1-59 (1975)) and B.R. (G. W. Beeler and H. Reuter, J. Physiol., Lond. 268, 177-210 (1977)) equations. The current mechanism iK2 has been replaced by the hyperpolarizing-activated current, i_f. Depletion and accumulation of potassium ions in the extracellular space are represented either by partial differential equations for diffusion in cylindrical or spherical preparations or, when such accuracy is not essential, by a three-compartment model in which the extracellular concentration in the intercellular space is uniform. The description of the delayed K current, i_K, remains based on the work of D. Noble and R. W. Tsien (J. Physiol., Lond. 200, 205-231 (1969a)). The instantaneous inward-rectifier, iK1, is based on S. Hagiwara and K. Takahashi's equation (J. Membrane Biol. 18, 61-80 (1974)) and on the patch clamp studies of B. Sakmann and G. Trube (J. Physiol., Lond. 347, 641-658 (1984)) and of Y. Momose, G. Szabo and W. R. Giles (Biophys. J. 41, 311a (1983)). The equations successfully account for all the properties formerly attributed to iK2, as well as giving more complete descriptions of iK1 and i_K. The sodium current equations are based on experimental data of T. J. Colatsky (J. Physiol., Lond. 305, 215-234 (1980)) and A. M. Brown, K. S. Lee and T. Powell (J. Physiol., Lond. 318, 479-500 (1981)). The equations correctly reproduce the range and magnitude of the sodium `window' current. The second inward current is based in part on the data of H. Reuter and H. Scholz (J. Physiol., Lond. 264, 17-47 (1977)) and K. S. Lee and R. W. Tsien (Nature, Lond. 297, 498-501 (1982)) so far as the ion selectivity is concerned. However, the activation and inactivation gating kinetics have been greatly speeded up to reproduce the very much faster currents recorded in recent work. A major consequence of this change is that Ca current inactivation mostly occurs very early in the action potential plateau. The sodium-potassium exchange pump equations are based on data reported by D. C. Gadsby (Proc. natn. Acad. Sci. U.S.A. 77, 4035-4039 (1980)) and by D. A. Eisner and W. J. Lederer (J. Physiol., Lond. 303, 441-474 (1980)). The sodium-calcium exchange current is based on L. J. Mullins' equations (J. gen. Physiol. 70, 681-695 (1977)). Intracellular calcium sequestration is represented by simple equations for uptake into a reticulum store which then reprimes a release store. The repriming equations use the data of W. R. Gibbons & H. A. Fozzard (J. gen. Physiol. 65, 367-384 (1975b)). Following Fabiato & Fabiato's work (J. Physiol., Lond. 249, 469-495 (1975)), Ca release is assumed to be triggered by intracellular free calcium. The equations reproduce the essential features of intracellular free calcium transients as measured with aequorin. The explanatory range of the model entirely includes and greatly extends that of the M.N.T. equations. Despite the major changes made, the overall time-course of the conductance changes to potassium ions strongly resembles that of the M.N.T. model. There are however important differences in the time courses of Na and Ca conductance changes. The Na conductance now includes a component due to the hyperpolarizing-activated current, i_f, which slowly increases during the pacemaker depolarization. The Ca conductance changes are very much faster than in the M.N.T. model so that in action potentials longer than about 50 ms the primary contribution of the fast gated calcium channel to the plateau is due to a steady-state `window' current or non-inactivated component. Slower calcium or Ca-activated currents, such as the Na-Ca exchange current, or Ca-gated currents, or a much slower Ca channel must then play the dynamic role previously attributed to the kinetics of a single type of calcium channel. This feature of the model in turn means that the repolarization process should be related to the inotropic state, as indicated by experimental work. The model successfully reproduces intracellular sodium concentration changes produced by variations in [Na]_o, or Na-K pump block. The sodium dependence of the overshoot potential is well reproduced despite the fact that steady state intracellular Na is proportional to extracellular Na, as in the experimental results of D. Ellis J. Physiol., Lond. 274, 211-240 (1977)). The model reproduces the responses to current pulses applied during the plateau and pacemaker phases. In particular, a substantial net decrease in conductance is predicted during the pacemaker depolarization despite the fact that the controlling process is an increase in conductance for the hyperpolarizing-activated current. The immediate effects of changing extracellular [K] are reproduced, including: (i) the shortening of action potential duration and suppression of pacemaker activity at high [K]; (ii) the increased automaticity at moderately low [K]; and (iii) the depolarization to the plateau range with premature depolarizations and low voltage oscillations at very low [K]. The ionic currents attributed to changes in Na-K pump activity are well reproduced. It is shown that the apparent K_m for K activation of the pump depends strongly on the size of the restricted extracellular space. With a 30% space (as in canine Purkinje fibres) the apparent K_m is close to the assumed real value of 1 mM. When the extracellular space is reduced to below 5%, the apparent K_m increases by up to an order of magnitude. A substantial part of the pump is then not available for inhibition by low [K]_b. These results can explain the apparent discrepancies in the literature concerning the K_m for pump activation.
Preview · Article · Feb 1985 · Philosophical Transactions of The Royal Society B Biological Sciences
[Show abstract][Hide abstract] ABSTRACT: 1. Voltage-clamp experiments on frog atrial muscle were designed to distinguish effects due to K accumulation in extracellular spaces from those due to activation of K conductance mechanisms in the membrane. 2. The set of instantaneous current-voltage relations obtained at various external K concentrations following depolarization to about -10 mV for several seconds was found to be quite different from that obtained before the depolarization. Hence the process of increasing the extracellular K concentration cannot account for all the time-dependent changes in outward current during depolarization. 3. Although the instantaneous current-voltage relations obtained at different values of external K concentration before prolonged depolarization show the cross-over phenomenon (Noble, 1965), those obtained at the end of the depolarization did not show this feature. It is concluded that the current-voltage relations for the channels conducting the time-dependent K current do not show cross-over. 4. These results were used to construct a model involving both K activation and K accumulation. This model successfully reproduces the appearance of a very slow component in outward current decay tails which, when subtracted by semi-exponential curve-stripping leaves a component with the real time constant of conductance change. The model does not however reproduce the appearance of a fast decaying component without adding a second conductance mechanism, or assuming non-exponential decay of a single conductance mechanism. 5. It is therefore suggested that i chi, fast is not a perturbation of i chi, slow or of iK1 by the process of K accumulation. This conclusion is reinforced by the results of experiments showing that the relative magnitude of i chi, fast is not greatly changed by substantially increasing the external K concentration in order to reduce the proportionate effect of K accumulation on the K concentration.
Preview · Article · Oct 1980 · The Journal of Physiology
[Show abstract][Hide abstract] ABSTRACT: 1. Regular perturbation theory was used to obtain analytical solutions for the time course of membrane current decay following voltage-clamp depolarizing pulses when both time-dependent K conductance mechanisms and the process of K accumulation in extracellular spaces are present. These solutions apply when the current and K concentration changes are small enough for linear relations to be assumed between current and K concentration. 2. In the case of a single Hodgkin-Huxley type conductance variable with time constant tau chi the presence of an accumulation process which, by itself, would produce a current decay with time constant tau alpha, induces the appearance of two infinite sets of components with decreasing time constants (1/(n+1/tau chi) and 1/(1/tau alpha + n/tau chi), where n is integer), and decreasing magnitudes. 3. The analytical solutions are used to investigate the range of conditions over which semi-exponential (curve-stripping) analysis of current decay tails may give useful information on the kinetics of current change. It is shown that, except at very large decay tail amplitudes, the method may give a good estimate of the true time constants of conductance decay even when the currents are assumed to be strongly dependent on external K concentration. 4. The method introduces error in current amplitude, but over the range in which curve-stripping gives useful results, the direct distortion of activation curves by variations in external K concentration is fairly small. However, as the current decay becomes grossly distorted in its time course by accumulation, so does the activation curve. The effects are very similar both to those obtained using numerical computation without linearization, and to those obtained experimentally. 5. Even with a large dependence of current on external K concentration the linear model does not reproduce i chi, fast as a perturbation of i chi, slow by K accumulation.
Preview · Article · Oct 1980 · The Journal of Physiology